Jumble Fail?

We have received much correspondence in the
last few days about Stephen's deck of cards in the 'Jumble' episode of series J. Was it shameful mathematics that led him to
claim that no pack in history had ever been in that same order?

Well here were the claims that Stephen
made:

'I am going to do something that has never
been done by any human being since the beginning of time.'

'That pack of cards has never before, in
the history of our planet been in that order'

'I can say with all the mathematical
certainty that is possible that this pack of cards has never been in this order
before. It's an absolute world first!'

It seems that the problem is the phrase
'mathematical certainty' which was said in the excitement of the moment and
which many people have (reasonably enough) taken as a reference to a pure
mathematical concept rather than a practical mathematical reality. However, generally speaking we stand by the
claims made on the show. We do believe
that nobody has ever had that exact shuffle before and we think that the maths
backs us up. It's possible that we're
wrong, but then it's also possible that if we etched a grain of sand with the
word QI and hid it randomly somewhere on the earth's beaches, you could stumble
across it at the first time of asking.
In fact, that feat would be positively commonplace compared with Stephen's
being a repeat of an earlier shuffle.

While we completely agree with the
multitudes of quibblers who point out that it is not impossible that Stephen's
shuffle had been done the day before, or the day after, or in fact any time
that anyone has shuffled a pack of cards, it is so vanishingly unlikely that
practically speaking we can confidently say that it has never happened before.

We tried to show how close to zero this
probability was by the following explanation:

'That number is so big that were you to
imagine that if every star in our galaxy had a trillion planets, each with a
trillion people living on them, and each of these people had a trillion pack of
cards, and somehow they managed to shuffle them all a thousand times a second,
and they'd been doing that since the big bang, they would only just now be
starting to repeat shuffles.'

Now the eagle-eyed mathematicians out there
might notice an error in there. In order
for this sentence to make sense, it is vitally important to point out that this
only holds if each shuffle is unique. Practically
speaking, having shuffled so many trillions and trillions of times, our
other-worldly shufflers would likely find some repetition. Repetition is unlikely, but as the number of
shuffles increases, it becomes inevitable.

To sum up: practically speaking we feel
that we were on safe ground in saying that Stephen's pack was shuffled in a way
that never been done by any human being since the beginning of time, but we
certainly could have been more elegant in our language. Probability is a slippery thing, in fact Bruno de Finetti, the Italian
probabilist, held that 'probability does not exist', rather that it is just a
subjective thing about how much you are prepared to bet on a certain
outcome.

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7 Comments

The probability shown was the probability of Stephen's shuffle being that exact shuffle. It was unrelaed to the probability of that particular shuffle never having been seen before.

This is somewhat similar to the birthday paradox. The odds of a particular random pair of people having the same birthday is astronomical, but the odds of there being two people in a group of people is much less (with only 22 people, there's an even chance that two of them will have the same birthday).

The math is beyond me, but I imagine that the chances of a particular shuffle _never_ having been seen before in the history of shuffles is probably even higher than that gigantic number. Or to put it another way, the odds are probably quite good that a particular shuffle has been shuffled before.

Did we miss the fact that Stephen very clearly said "THAT pack of cards"? For all we know, that particular deck had never been shuffled before that moment. Even if it had been shuffled hundreds of times, as noted in other comments the resulting card order is almost certainly new FOR THAT DECK.

Do you have any evidence that card packs come in a similar order from a factory? My personal experience is that they don't (apart from the jokers and the bridge card). In my experience they come in a "pre shuffled form" (random order) so that you can get playing with them as quickly as possible.

The mathematics only says that over time, the frequency of particular sequence of cards appearing will approach once in every 52! times. It says nothing about what has gone before, nor what has gone after. It most certainly does not say that the sequence will occur exactly once in 52! shuffles.

As the de Finetti quote suggests, the odds only affect how much you're willing to bet on the next shuffle. It should also be added that if you're going to bet on a truly random event, consideration of anything that has gone before is truly irrational.